A standard theorem says that any locally convex topological vector
space has a finer topology, its Mackey topology with the same
set of continuous linear functionals and that is the finest possible
topology with that property. If E and F are two such spaces,
topologize the space Hom(E,F) of continuous linear transformations
E®F with the weak topology induced by the algebraic tensor
product EÄF¢ and then let [E,F] denote the associated
Mackey topology. Let F* denote the dual F¢ topologized by the
Mackey topology on the weak dual and let EÄF=[E,F*]* (whose
underlying vector space is the algebraic tensor product). Then for
any Mackey spaces E, F, and G,

[EÄF,G] @ [E,[F,G]]

E@E**

[E,F] @ (EÄF*)*

which is summarized by saying that the category of Mackey spaces and
continuous linear transformations is *-autonomous.

This category is equivalent to the category of weakly topologized
locally convex topological vector spaces (which have the coarsest
possible topology for their set of continuous linear functionals)
which is therefore also *-autonomous. They are also equivalent to
the chu category of vector spaces (which will be explained).

The motivational example for the comprehension scheme (Lawvere '68)
came from proof theory. An example with categories as types (Gray
'69, Street-Walters '73) exhibited comprehension as the familiar
Grothendieck construction of a discrete opfibration associated with a
covariant functor F : B® Sets on a small category B.

We introduce the setting of an "extensive 2-doctrine" (E2D) in which
to state the comprehension scheme in a 2-categorical setting. This
involves a 2-category T "of types" and, for each object X of
T, a category E(X) of "extensive quantities of type X" with a
terminal object 1X, and a "pushforward operation" E(f) :E(Y) ®E(X) for each 1-cell f : Y®X in T. For each
object X of T, we have a 2-functor BX : (T,X) ®E(X)
that assigns, to each 1-cell f : Y®X, the extensive
quantity of type X given by E(f)(1Y). We say that the E2D
satisfies the comprehension scheme if for each X, the 2-functor
BX has a fully faithful right 2-adjoint {-}B : E(X) ® (T,X), called comprehension. A 1-cell f : Y®X is called
E-dense if the canonical map E(f)(1Y) ® 1X is an isomorphism,
and it is called an E-covering if the unit f® {BX (f)}X is an
isomorphism. It follows that every 1-cell f : Y®X admits
a unique (up to iso) factorization into an E-dense 1-cell Y®Z, followed by an E-covering 1-cell Z®X. This is called
the "E-comprehensive factorization" of f.

The purpose of this talk is:

(1) to remark that the (pure, Fox complete spread)
factorization (Bunge-Funk '96) is indeed comprehensive for an E2D
with T the 2-category of locally connected (Grothendieck) toposes
and E(X) the category of Lawvere distributions on X, and
(2) to prove that the (hyperpure, Michael Complete spread)
factorization (Bunge-Funk 2005) is comprehensive for an E2D with T
the 2-category of all (Grothendieck) toposes and E(X) a category of
what we call "0-distributions", or distributions with values in
0-dimensional locales.

The relation R* is defined on a monoid M by the rule
that aR*b if and only if for all x,yÎM,

xa = yaÛxb = yb.

If the set of idempotents E(M) of M is a commutative submonoid of
M and every R*-class contains an idempotent, M is
said to be left adequate. In such a monoid each
R*-class contains a unique idempotent and the idempotent
in the R*-class of an element a is denoted by
af. A left adequate monoid M is left ample if ae = (ae)fa for all eÎE(M) and aÎM.

Thus a right cancellative monoid is left ample; here R*
is the universal relation. Every inverse monoid is left ample.

A left ample monoid is proper if the intersection of the
minimum left cancellative congruence and R* is trivial.
The structure of proper left ample monoids can be described in terms
of right cancellative monoids and commutative monoids of idempotents.
Moreover, any left ample monoid M has a proper cover, that
is, a proper left ample monoid P together with a homomorphism from
P onto M which restricts to an isomorphism from E(P) onto
E(M). We consider how such covers can be constructed.

Some anciently studied categories turn out to have *-autonomous
structures not previously noted; for example, the category of finitely
generated additive group-valued functors from finitely generated
abelian groups. This is, in fact, the free abelian category on one
object generator. As all free structures on one generator it has a
monoidal structure (it may be identified as composition of functors).
It is neither the tensor product nor the "par" but lies between
them.

We shall present a strictly semigroup description of the classifying
topos B(G) [2] of an inverse semigroup G. A
left *-semigroup is a semigroup S together with an assignment
s®s* satisfying:

(i) (s*)* = s,
(ii) ss*s = s, and
(iii) (s*st)* = (st)*s,
for all s,tÎS. A morphism of left *-semigroups is a function
h : S®T such that
(i) h(s*) = h(s)*, and
(ii) h(st) = h(s)h(s*st).
Such a morphism h is said to be étale if every equation
t=h(f)t in T, where f is a strong idempotent (f=f*f) of S,
can be lifted uniquely to an equation s=fs in S, meaning
h(s)=t.

Proposition 1B(G) is equivalent to the category of étale morphisms of
left *-semigroups over the inverse semigroup G.

We shall also present a strictly topos description of E-unitary
inverse semigroups [3]. A ØØ-separated object
of a topos is one that is separated for the ØØ-topology in the
topos [1]. (F is ØØ-separated iff the diagonal
subobject F\rightarrowtail F×F is equal to its double
negation.)

Proposition 2
An inverse semigroup G is E-unitary iff the object d :G®E of B(G) is ØØ-separated, where E=
idempotents of G, and d(t) = t*t.

A monoidal category is equipped with associativity and unit
isomorphisms, subject to coherence conditions. Under a
commonly-satisfied hypothesis ("tensor generation"), these coherence
conditions are redundant-in the sense that the existence of
arbitrary associativity and unit isomorphisms implies the existence of
a coherent collection of them. The same principle extends to the
symmetric case.

I shall explain the hypothesis, and describe how (when the hypothesis
is satisfied) coherent associativity, symmetry, and unit isomorphisms
may be constructed from arbitrary ones.

The theory of quasi-categories is extending both category theory and
homotopy theory. We shall discuss the similarities and the
differences between these theories. We shall discuss the somewhat
surprising fact that a general groupoid can be treated as an
equivalence relation. In particular, a group defines an equivalence
relation on a point; the quotient is the classifying space of the
group.

Given a category C with a suitable class M of morphisms, one
obtains a corresponding notion of partial map, where the M's provide
the possible domains of definition. Typically the M's will be
monomorphisms.

If C has higher-dimensional structure (for example if it is a
bicategory or tricategory) then new phenomena occur; among other
things, one might expect to interpret the condition that the M's be
monomorphisms in a relaxed way suitable for a bicategory or
tricategory.

In this talk I will focus on the case where C is the (2-)category of
categories and where C is a certain category of bicategories. In
particular I will describe how "partial morphisms of bicategories"
are the same thing as the "2-sided enrichments" of Kelly, Labella,
Schmitt, and Street.

In the 1960s, Richard Thompson (and, independently, Freyd and Heller)
discovered three groups, F, T and V, with several remarkable
properties. F, in particular, turns out to be one of those
structures that appears unexpectedly in many diverse parts of
mathematics. It also has a very natural and simple categorical
description: it is the symmetry group of the `generic idempotent
object'. I will explain what this means, how it differs from Freyd
and Heller's earlier description, and how it belongs to the large
family of existing descriptions of free categories with structure.

The P2 construction [2] freely adds right adjoints to all arrows
(or a suitably chosen subset of the arrows) of a category. When one
considers this construction, one may wonder what one needs to do in
order to obtain a 2-category in which every arrow has both a left and
a right adjoint. As was shown in [1], such arrows come in cycles,
either an infinite cycle or a finite cycle. In this talk we will
present a construction which freely adds specified cycles of adjoints
to classes of arrows in a category that is freely generated on a
graph. (This is our first step toward such a construction for
arbitrary categories and 2-categories.) As a result we obtain
families of non-trivial examples of categories containing cycles of
arrows which are both left and right adjoints.

Quantales are simple algebraic structures which can be found very
often, explicitly or less so, in mathematics. They have properties
that make them analogous to rings, and similarly to rings the richer
aspects of the theory only become available when we restrict to
quantales satisfying special conditions.

In this talk we shall mainly address a class of quantales that is both
easy to describe and closely related to groupoids and inverse
semigroups, namely the so-called inverse quantale frames, which form a
category that is equivalent to the category of complete and infinitely
distributive inverse semigroups. Partly because of this equivalence,
quantales turn out to be good mediating objects for the purpose of
constructing étale groupoids from inverse semigroups (for instance
the germ groupoid of a pseudogroup, or Paterson's universal groupoid
of an inverse semigroup).

We shall examine the three-fold interplay between quantales, groupoids
and inverse semigroups, and some of its known or conjectured
consequences as regards one or more of the following topics: more
general semigroups (such as guarded semigroups); more general
groupoids (such as open groupoids); generalizations of groupoid
cohomology; the structure of groupoid C*-algebras.

Girard's Geometry of Interaction (GoI) program develops a mathematical
modelling of the dynamics of cut-elimination in proof-theory.
Girard's work (1988-1995, 2004-) is stated in the language of
operator algebras. He gave a novel modelling of proofs, interpreting
cuts via feedback in an intrinsic theory of types, data and
algorithms. However, as emphasized by Hyland and Abramsky, there are
deep connections of GoI with the recent theory of traced monoidal
categories of Joyal-Street-Verity. Indeed, traces lead to new
insights into Girard's Execution Formula, a kind of power series
representing an invariant of cut-elimination. Recently, in a series
of papers, E. Haghverdi and I have re-examined the categorical
foundations of GoI. For example, we develop a typed version,
Multiobject GoI (MGoI), which includes all previous as well as several
new models. MGoI depends on a new theory of partial traces, trace
classes and an abstract theory of orthogonality (related to work of
Hyland and Schalk). I shall survey some of this recent work, along
with Soundness and Completeness Theorems for GoI semantics. If time
permits, we also explore some of the new directions in GoI.

We introduce the notion of a differential category: a
(semi-)additive symmetric monoidal category with a comonad (a
"coalgebra modality") and a differential combinator, satisfying a
number of coherence conditions. In such a category, one should regard
the base maps as "linear", and the coKleisli maps as "smooth"
(infinitely differentiable). Although such categories do not
necessarily arise from models of linear logic, one should think of
this as replacing the usual dichotomy of linear vs. stable
maps established for coherence spaces.

To illustrate this approach, we give a number of examples, the most
important of which, a monad S¥ on the category of vector
spaces, with a canonical differential combinator, fully captures the
usual notion of derivatives of smooth maps. Our models are somewhat
more general than are allowed by other approaches (such as Ehrhard's
and Regnier's, which inspired our work). For example, differential
categories are monoidal categories, rather than monoidal closed or
*-autonomous categories. This allows us to capture various
"standard models" of differentiation which are notably not closed.
Second, we relax the condition that the comonad be a "storage"
modality in the usual sense of linear logic, again so as to allow the
standard models which do not necessarily give rise to a full storage
modality. However, when the comonad is a storage modality, we can
describe an extension of the notion of differential category which
captures the not-necessarily-closed fragment of Ehrhard-Regnier's
differential l-calculus.

Algebraic set theory considers the category-theoretic structure
implicit in the notion of smallness arising from the set/class size
distinction of first-order set theory. Hitherto, it has mainly
aroused foundational interest as a reorganization of models of (many
variants of) set theory, focusing on their algebraic structure.
However, by providing a natural environment for carrying out internal
arguments involving large structures and size distinctions, algebraic
set theory should also be a theory with applications. In this talk, I
shall attempt to explore some of the possible directions such
applications might take.

Using Mobius inversion, we give an explicit isomorphism between the
algebra of a finite inverse semigroup and the algebra of its
underlying groupoid (which is in turn isomorphic to a direct sum of
matrix algebras over the local groups). From this one obtains a
description of the irreducible representations and a character sum
formula for calculating intertwining numbers. Applications include
explicit decompositions of tensor and exterior powers of
representations of partial permutation inverse semigroups and
calculation of the eigenvalues with multiplicities for random walks on
finite triangularizable semigroups.

The many schools of computable or constructive analysis accept without
question the received notion of set with structure. They rein in the
wild behaviour of set-theoretic functions using the double bridle of
topology and recursion theory, adding encodings of explicit numerical
representations to the epsilons and deltas of metrical analysis.
Fundamental conceptual results such as the Heine-Borel theorem can
only be saved by set-theoretic tricks such as Turing tapes with
infinitely many non-trivial symbols.

It doesn't have to be like that.

When studying computable continuous functions, we should never
consider uncomputable or discontinuous ones, only to exclude them
later. By the analogy between topology and computation, we
concentrate on open subspaces. So we admit +, -, ×,
¸, < , > , ¹ , Ù and Ú, but not £ , ³ ,
=, \lnot or Þ. Universal quantification captures the
Heine-Borel theorem, being allowed over compact spaces.
Dedekind completeness can also be presented in a natural logical style
that is much simpler than the constructive notion of Cauchy sequence,
and also more natural for both analysis and computation.

Since open subspaces are defined as continuous functions to the
Sierpi\'nski space, rather than as subsets, they enjoy a "de Morgan"
duality with closed subspaces that is lost in intuitionistic set-,
type- or topos theories. Dual to " compact spaces is
$ over "overt" spaces. Classically, all spaces are overt,
whilst other constructive theories use explicit enumerations or
distance functions instead. Arguments using $ and overtness
are both dramatically simpler and formally dual to familiar ideas
about compactness.

Recently, in two unpublished papers, Streicher and Lubarsky have
(independently) put forward realisability models for CZF. In this
talk, I will show that the two models are the same and make clear how
their work can be understood in the context of algebraic set theory
(AST). I hope also to indicate how AST can be used to demonstrate the
validity of several principles in this model. (This is all part of the
speaker's PhD thesis.)

It is shown how David McCarty's well-known realizability model for IZF
fits into Joyal and Moerdijk's framework of Algebraic Set Theory. We
shall remark on a way to eliminate the external ordinals from the
construction, as well as using recent work of M. Warren in order to
obtain proof-theoretic properties of the model by topos-theoretic
means.

It is a basic result from topos theory that if E is an
elementary topos and G is a cartesian comonad on E, then
the category EG of coalgebras for G is also an
elementary topos. We extend this result to the setting of algebraic
set theory by showing that if C is one of several kinds of
categories of classes and G is a cartesian comonad which preserves
small maps, then CG is also a category of classes of the
same kind as C. We then turn to the consideration of
several useful corollaries. First, categories of classes are, under
suitable conditions, stable under the formation of internal
presheaves. Secondly, it follows that several of the set theories
considered in the literature on algebraic set theory possess the
disjunction and existence properties.

The parametrized 2-category constructions Fib/S, for S with finite
limits, and W-cat, for W a bicategory, are further unified by
considering, for fixed W, the 2-category of pseudo-functors H: A®W which are locally discrete fibrations. This
2-category is biequivalently described as a 2-category whose objects
are lax-functors Wco® mat, where mat is the bicategory
whose objects are sets and whose hom-categories are given by
mat(X,A) = setAxX. The biequivalence is a direct
generalization of the Grothendieck biequivalence between fibrations
and CAT-valued pseudo-functors and is mediated by pulling back a
universal local discrete fibration mat*® mat. Further, the
2-category is also biequivalent to the classical [^(]W)-cat, where
[^(]W) is the bicategory whose objects are those of W with
[^(]W)(w,x) = setW(w,x)op. We will show how to recover the
usual variable and enriched categories within this framework.